On algebraic damping close to inhomogeneous Vlasov equilibria in multi-dimensional spaces

TL;DR

This paper analyzes how perturbations decay over time in multi-dimensional Vlasov systems, revealing algebraic damping caused by singularities in the Fourier-Laplace transform, with classifications and computations in 2D systems.

Contribution

It provides a detailed classification of singularities and damping rates for inhomogeneous Vlasov equilibria in multi-dimensional spaces, extending understanding of damping mechanisms.

Findings

Algebraic damping arises from branch singularities in the Fourier-Laplace transform.

Classification of singularities and damping rates in two-dimensional systems.

Validation of the theory using toy models and the isochrone model.

Abstract

We investigate the asymptotic damping of a perturbation around inhomogeneous stable stationary states of the Vlasov equation in spatially multi-dimensional systems. We show that branch singularities of the Fourier-Laplace transform of the perturbation yield algebraic dampings. In two spatial dimensions, we classify the singularities and compute the associated damping rate and frequency. This 2D setting also applies to spherically symmetric self-gravitating systems. We validate the theory using a toy model and an advection equation associated with the isochrone model, a model of spherical self-gravitating systems.